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Chapter 2 Equilibrium Chapter 2 Equilibrium The theory of equilibrium attempts to predict what happens in a game when players be- have strategically. This is a central concept to this text as, in mechanism design, we are optimizing over games to find the games with good equilibria. Here, we review the most fundamental notions of equilibrium. They will all be static notions in that players are as- sumed to understand the game and will play once in the game. While such foreknowledge is certainly questionable, some justification can be derived from imagining the game in a dynamic setting where players can learn from past play. Readers should look elsewhere for formal justifications. This chapter reviews equilibrium in both complete and incomplete information games. As games of incomplete information are the most central to mechanism design, special at- tention will be paid to them. In particular, we will characterize equilibrium when the private information of each agent is single-dimensional and corresponds, for instance, to a value for receiving a good or service. We will show that auctions with the same equilibrium outcome have the same expected revenue. Using this so-called revenue equivalence we will describe how to solve for the equilibrium strategies of standard auctions in symmetric environments. Emphasis is placed on demonstrating the central theories of equilibrium and not on providing the most comprehensive or general results. For that readers are recommended to consult a game theory textbook. 2.1 Complete Information Games In games of compete information all players are assumed to know precisely the payoff struc- ture of all other players for all possible outcomes of the game. A classic example of such a game is the prisoner’s dilemma, the story for which is as follows. Two prisoners who have jointly committed a crime, are being interrogated in separate quarters. Unfortunately, the interrogators are unable to prosecute either prisoner without a confession. Each prisoner is offered the following deal: If she confesses and their accomplice does not, she will be released and her accomplice will serve the full sentence of ten years in prison. If they both confess, they will 25 share the sentence and serve five years each. If neither confesses, they will both be prosecuted for a minimal offense and each receive a year of prison. This story can be expressed as the following bimatrix game where entry (a, b) represents row player’s payoff a and column player’s payoff b. silent confess silent (-1,-1) (-10,0) confess (0,-10) (-5,-5) A simple thought experiment enables prediction of what will happen in the prisoners’ dilemma. Suppose the row player is silent. What should the column player do? Remaining silent as well results in one year of prison while confessing results in immediate release. Clearly confessing is better. Now suppose that the row player confesses. Now what should the column player do? Remaining silent results in ten years of prison while confessing as well results in only five. Clearly confessing is better. In other words, no matter what the row player does, the column player is better of by confessing. The prisoners dilemma is hardly a dilemma at all: the strategy profile (confess, confess) is a dominant strategy equilibrium. Definition 2.1. A dominant strategy equilibrium (DSE) in a complete information game is a strategy profile in which each player’s strategy is as least as good as all other strategies regardless of the strategies of all other players. DSE is a strong notion of equilibrium and is therefore unsurprisingly rare. For an equi- librium notion to be complete it should identify equilibrium in every game. Another well studied game is chicken. James Dean and Buzz (in the movie Rebel without a Cause) face off at opposite ends of the street. On the signal they race their cars on a collision course towards each other. The options each have are to swerve or to stay their course. Clearly if they both stay their course they crash. If they both swerve (opposite directions) they escape with their lives but the match is a draw. Finally, if one swerves and the other stays, the one that stays is the victor and the other the loses.1 A reasonable bimatrix game depicting this story is the following. stay swerve stay (-10,-10) (1,-1) swerve (-1,1) (0,0) Again, a simple thought experiment enables us to predict how the players might play. Suppose James Dean is going to stay, what should Buzz do? If Buzz stays they crash and Buzz’s payoff is −10, but if Buzz swerves his payoff is only −1. Clearly, of these two options 1The actual chicken game depicted in Rebel without a Cause is slightly different from the one described here. 26 Buzz prefers to swerve. Suppose now that Buzz is going to swerve, what should James Dean do? If James Dean stays he wins and his payoff is 1, but if he swerves it is a draw and his payoff is zero. Clearly, of these two options James Dean prefers to stay. What we have shown is that the strategy profile (stay, swerve) is a mutual best response, a.k.a., a Nash equilibrium. Definition 2.2. A Nash equilibrium in a game of complete information is a strategy profile where each players strategy is a best response to the strategies of the other players as given by the strategy profile. In the examples above, the strategies of the players correspond directly to actions in the game, a.k.a., pure strategies. In general, Nash equilibrium strategies can be randomizations over actions in the game, a.k.a., mixed strategies. 2.2 Incomplete Information Games Now we turn to the case where the payoff structure of the game is not completely known. We will assume that each agent has some private information and this information affects the payoff of this agent in the game. We will refer to this information as the agent’s type and denote it by ti for agent i. The profile of types for the n agents in the game is t =(t1,...,tn). A strategy in a game of incomplete information is a function that maps an agent’s type to any of the agent’s possible actions in the game (or a distribution over actions for mixed strategies). We will denote by si(·) the strategy of agent i and s = (s1,...,sn) a strategy profile. The auctions described in Chapter 1 were games of incomplete information where an agent’s private type was her value for receiving the item, i.e., ti = vi. As we described, strategies in the English auction were si(vi) = “drop out when the price exceeds vi” and strategies in the second-price auction were si(vi) = “bid bi = vi.” We refer to this latter strategy as truthtelling. Both of these strategy profiles are in dominant strategy equilibrium for their respective games. Definition 2.3. A dominant strategy equilibrium (DSE) is a strategy profile s such that for all i, ti, and b−i (where b−i generically refers to the actions of all players but i), agent i’s utility is maximized by following strategy si(ti). Notice that aside from strategies being defined as a map from types to actions, this definition of DSE is identical to the definition of DSE for games of complete information. 2.3 Bayes-Nash Equilibrium Naturally, many games of incomplete information do not have dominant strategy equilibria. Therefore, we will also need to generalize Nash equilibrium to this setting. Recall that equilibrium is a property of a strategy profile. It is in equilibrium if each agent does not 27 want to change her strategy given the other agents’ strategies. Meaning, for an agent i, we want to the fix other agent strategies and let i optimize her strategy (meaning: calculate her best response for all possible types ti she may have). This is an ill specified optimization as just knowing the other agents’ strategies is not enough to calculate a best response. Additionally, i’s best response depends additionally on i’s beliefs on the types of the other agents. The standard economic treatment addresses this by assuming a common prior. Definition 2.4. Under the common prior assumption, the agent types t are drawn at random from a prior distribution F (a joint probability distribution over type profiles) and this prior distribution is common knowledge. The distribution F over t may generally be correlated. Which means that an agent with knowledge of her own type must do Bayesian updating to determine the distribution over the types of the remaining bidders. We denote this conditional distribution as F− . Of course, i ti when the distribution of types is independent, i.e., F is the product distribution F ×···×F , 1 n then F− = F− . i ti i Notice that a prior F and strategies s induces a distribution over the actions of each of the agents. With such a distribution over actions, the problem each agent faces of optimizing her own action is fully specified. Definition 2.5. A Bayes-Nash equilibrium (BNE) for a game G and common prior F is a strategy profile s such that for all i and ti, si(ti) is a best response when other agents play s− (t− ) when t− ∼ F− . i i i i ti To illustrate BNE, consider using the first-price auction to sell a single item to one of two agents, each with valuation drawn independently and identically from U[0, 1], i.e., F = F ×F with F (z) = Prv∼F [v < z] = z.
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